Chapter 6

Solve each inequality. $$\frac{x+1}{x-2}<1$$

Set up an equation and solve each problem. Charlotte's time to travel 250 miles is 1 hour more than Lorraine's time to travel 180 miles. Charlotte drove 5 miles per hour faster than Lorraine. How fast did each one travel?

Expressing solutions to the nearest one-thousandth. $$x^{2}-16 x-24=0$$

Solve each quadratic equation using the method that seems most appropriate. $$4 x^{2}-8 x+3=0$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\frac{\sqrt{-25}}{\sqrt{-4}}$$

Solve each inequality. $$\frac{x+3}{x-4} \geq 1$$

Set up an equation and solve each problem. Larry's time to travel 156 miles is 1 hour more than Terrell's time to travel 108 miles. Terrell drove 2 miles per hour faster than Larry. How fast did each one travel?

Expressing solutions to the nearest one-thousandth. $$x^{2}+6 x-44=0$$

Solve each quadratic equation using the method that seems most appropriate. $$9 x^{2}+18 x+5=0$$

Use Property \(6.1\) to help solve each quadratic equation. $$(x-2)^{2}=49$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\frac{\sqrt{-81}}{\sqrt{-9}}$$

Explain how to solve the inequality \((x+1)(x-2)\) \((x-3)>0\).

Set up an equation and solve each problem. On a 570 -mile trip, Andy averaged 5 miles per hour faster for the last 240 miles than he did for the first 330 miles. The entire trip took 10 hours. How fast did he travel for the first 330 miles?

Expressing solutions to the nearest one-thousandth. $$x^{2}+10 x-46=0$$

Solve each quadratic equation using the method that seems most appropriate. $$x^{2}+12 x=4$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\frac{\sqrt{-56}}{\sqrt{-7}}$$

Explain how to solve the inequality \((x-2)^{2}>0\) by inspection.

Set up an equation and solve each problem. On a 135 -mile bicycle excursion, Maria averaged 5 miles per hour faster for the first 60 miles than she did for the last 75 miles. The entire trip took 8 hours. Find her rate for the first 60 miles.

Expressing solutions to the nearest one-thousandth. $$x^{2}+8 x+2=0$$

Solve each quadratic equation using the method that seems most appropriate. $$x^{2}+6 x=-11$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\frac{\sqrt{-72}}{\sqrt{-6}}$$

Your friend looks at the inequality \(1+\frac{1}{x}>2\) and without any computation states that the solution set is all real numbers between 0 and 1 . How can she do that?

Set up an equation and solve each problem. It takes Terry 2 hours longer to do a certain job than it takes Tom. They worked together for 3 hours; then Tom left and Terry finished the job in 1 hour. How long would it take each of them to do the job alone?

Expressing solutions to the nearest one-thousandth. $$x^{2}+9 x+3=0$$

Solve each quadratic equation using the method that seems most appropriate. $$4(2 x+1)^{2}-1=11$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\frac{\sqrt{-24}}{\sqrt{6}}$$

Why is the solution set for \((x-2)^{2} \geq 0\) the set of all real numbers?

Set up an equation and solve each problem. Suppose that Arlene can mow the entire lawn in 40 minutes less time with the power mower than she can with the push mower. One day the power mower broke down after she had been mowing for 30 minutes. She finished the lawn with the push mower in 20 minutes. How long does it take Arlene to mow the entire lawn with the power mower?

Expressing solutions to the nearest one-thousandth. $$4 x^{2}-6 x+1=0$$

Solve each quadratic equation using the method that seems most appropriate. $$5(x+2)^{2}+1=16$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\frac{\sqrt{-96}}{\sqrt{2}}$$

Set up an equation and solve each problem. A student did a word processing job for \(\$ 24\). It took him 1 hour longer than he expected, and therefore he earned \(\$ 4\) per hour less than he anticipated. How long did he expect that it would take to do the job?

Expressing solutions to the nearest one-thousandth. $$5 x^{2}-9 x+1=0$$

Use the method of completing the square to solve \(a x^{2}+\) \(b x+c=0\) for \(x\), where \(a, b\), and \(c\) are real numbers and \(a \neq 0\).

Find each of the products and express the answers in the standard form of a complex number. $$(5 i)(4 i)$$

The product \((x-2)(x+3)\) is positive if both factors are negative or if both factors are positive. Therefore, we can solve \((x-2)(x+3)>0\) as follows: $$ \begin{gathered} (x-2<0 \text { and } x+3<0) \text { or }(x-2>0 \text { and } x+3>0) \\ (x<2 \text { and } x<-3) \text { or }(x>2 \text { and } x>-3) \\ x<-3 \text { or } x>2 \end{gathered} $$ The solution set is \((-\infty,-3) \cup(2, \infty)\). Use this type of analysis to solve each of the following. (a) \((x-2)(x+7)>0\) (b) \((x-3)(x+9) \geq 0\) (c) \((x+1)(x-6) \leq 0\) (d) \((x+4)(x-8)<0\) (e) \(\frac{x+4}{x-7}>0\) (f) \(\frac{x-5}{x+8} \leq 0\)

Set up an equation and solve each problem. A group of students agreed that each would chip in the same amount to pay for a party that would cost \(\$ 100\). Then they found 5 more students interested in the party and in sharing the expenses. This decreased the amount each had to pay by \(\$ 1\). How many students were involved in the party and how much did each student have to pay?

Expressing solutions to the nearest one-thousandth. $$2 x^{2}-11 x-5=0$$

Find each of the products and express the answers in the standard form of a complex number. $$(-6 i)(9 i)$$

Set up an equation and solve each problem. A group of students agreed that each would contribute the same amount to buy their favorite teacher an \(\$ 80\) birthday gift. At the last minute, 2 of the students decided not to chip in. This increased the amount that the remaining students had to pay by \(\$ 2\) per student. How many students actually contributed to the gift?

Expressing solutions to the nearest one-thousandth. $$3 x^{2}-12 x-10=0$$

Give a step-by-step description of how to solve \(3 x^{2}+9 x-\) \(4=0\) by completing the square.

Find each of the products and express the answers in the standard form of a complex number. $$(7 i)(-6 i)$$

Set up an equation and solve each problem. A retailer bought a number of special mugs for \(\$ 48\). She decided to keep two of the mugs for herself but then had to change the price to \(\$ 3\) a mug above the original cost per mug. If she sells the remaining mugs for \(\$ 70\), how many mugs did she buy and at what price per mug did she sell them?

Use the discriminant to help solve each problem. Determine \(k\) so that the solutions of \(x^{2}-2 x+k=0\) are complex but nonreal.

For the indicated variable. Assume that all letters represent positive numbers. \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) for \(y\)

Find each of the products and express the answers in the standard form of a complex number. $$(-5 i)(-12 i)$$

Use the discriminant to help solve each problem. Determine \(k\) so that \(4 x^{2}-k x+1=0\) has two equal real solutions.

For the indicated variable. Assume that all letters represent positive numbers. \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) for \(x\)

Find each of the products and express the answers in the standard form of a complex number. $$(3 i)(2-5 i)$$